Dorothee D. Haroske

Envelopes and sharp embeddings of function spaces

Preface DEFINITION, BASIC PROPERTIES, AND FIRST EXAMPLES Introduction Preliminaries, Classical Function Spaces Non-increasing rearrangements Lebesgue and Lorentz spaces Spaces of continuous functions Sobolev spaces Sobolevs embedding theorem The Growth Envelope Function EG Definition and basic properties Examples: Lorentz spaces Connection with the fundamental function Further examples: Sobolev spaces, weighted Lp-spaces Growth Envelopes EG Definition Examples: Lorentz spaces, Sobolev spaces The Continuity Envelope Function EC Definition and basic properties Some lift property Examples: Lipschitz spaces, Sobolev spaces Continuity Envelopes EC Definition Examples: Lipschitz spaces, Sobolev spaces RESULTS IN FUNCTION SPACES AND APPLICATIONS Function Spaces and Embeddings Spaces of type Bsp,q, Fsp,q Embeddings Growth Envelopes EG Growth envelopes in the sub-critical case Growth envelopes in sub-critical borderline cases Growth envelopes in the critical case Continuity Envelopes EC Continuity envelopes in the super-critical case Continuity envelopes in the super-critical borderline case Continuity envelopes in the critical case Envelope Functions EG and EC Revisited Spaces on R+ Enveloping functions Global versus local assertions Applications Hardy inequalities and limiting embeddings Envelopes and lifts Compact embeddings References Symbols Index List of Figures

Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers

We study continuity envelopes in spaces of generalised smoothness Bpq(s,Ψ) and Fpq(s,Ψ) and give some new characterisations for spaces Bpq(s,Ψ) The results are applied to obtain sharp asymptotic estimates for approximation numbers of compact embeddings of type id: Bpq(s1,Ψ) (U) → Bx-xs2 (U), Where n/p < s1-s2 < n/p+1 and U stands for the unit ball in Rn In case of entropy numbers we can prove two-sided estimates.

Entropy numbers of embeddings of function spaces with Muckenhoupt weights, III. Some limiting cases

We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt Ap class. This extends our previous results [25] to more general weights of logarithmically disturbed polynomial growth, both near some singular point and at infinity. We obtain sharp asymptotic estimates for the entropy numbers of this embedding. Essential tools are a discretisation in terms of wavelet bases, as well as a refined study of associated embeddings in sequence spaces and interpolation arguments in endpoint situations.

On trace spaces of function spaces with a radial weight: the atomic approach

We study Besov and Triebel–Lizorkin spaces with a special Muckenhoupt weight of type w(x) ∼ |x|α and determine their trace spaces with respect to the hyperplane ℝ n−1. The approach is based on atomic decomposition results.

Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers

Continuity envelopes for the spaces of generalised smoothness Bpq(s,Ψ)(ℝn) and Fpq(s,Ψ)(ℝn) are studied in the so-called supercritical s=1

Estimates for continuity envelopes and approximation numbers of Bessel potentials

Abstract In this paper we study spaces of Bessel potentials in n -dimensional Euclidean spaces. They are constructed on the basis of a rearrangement-invariant space (RIS) by using convolutions with Bessel–MacDonald kernels. Specifically, the treatment covers spaces of classical Bessel potentials. We establish two-sided estimates for the corresponding modulus of smoothness of order k ∈ N , ω k ( f ; t ) , and determine their continuity envelope functions. This result is then applied to estimate the approximation numbers of some embeddings.

On More General Lipschitz Spaces

The present paper deals with (logarithmic) Lipschitz spaces of type Lip(1,−α) p,q (1 ≤ p ≤ ∞, 0 < q ≤ ∞, α > 1 q ). We study their properties and derive some (sharp) embedding results. In that sense this paper can be regarded as some continuation and extension of our papers [8, 9], but there are also connections with some recent work of Triebel concerning Hardy inequalities and sharp embeddings. Recall that the nowadays almost ‘classical’ forerunner of investigations of this type is the Brézis-Wainger result [6] about the ‘almost’ Lipschitz continuity of elements of the Sobolev spaces H 1+ n p p (R) when 1 < p < ∞.

Embedding properties of weighted Besov-type spaces

In this paper, we consider the embeddings of weighted Besov spaces into Besov-type spaces with w being a (local) Muckenhoupt weight, and give sufficient and necessary conditions on the continuity and the compactness of these embeddings. As special cases, we characterize the continuity and the compactness of embeddings in case of some polynomial or exponential weights. The proofs of these conclusions strongly depend on the geometric properties of dyadic cubes.

Limiting embeddings in smoothness Morrey spaces, continuity envelopes and applications

In this paper, the authors prove some Franke-Jawerth embedding for the Besov-type spaces B p , q s , ? ( R n ) and the Triebel-Lizorkin-type spaces F p , q s , ? ( R n ) . By using some limiting embedding properties of these spaces and the Besov-Morrey spaces N u , p , q s ( R n ) , the continuity envelopes in B p , q s , ? ( R n ) , F p , q s , ? ( R n ) and N u , p , q s ( R n ) are also worked out. As applications, the authors present some Hardy type inequalities in the scales of B p , q s , ? ( R n ) , F p , q s , ? ( R n ) and N u , p , q s ( R n ) , and also give the estimates for approximation numbers of the embeddings from B p , q s , ? ( ? ) , F p , q s , ? ( ? ) and N u , p , q s ( ? ) into C ( ? ) , where ? denotes the unit ball in R n .

Embedding properties of Besov-type spaces

In this paper, the authors study embeddings of Besov-type spaces , and obtain necessary and sufficient conditions. Moreover, the authors also consider situations or , and can finally cover some cases where is replaced by .